We remark here that it is not possible to expect \(|B^{\prime }+cB^{\prime }|\le |B^{\prime }|^{2-\epsilon }\)  for any \(\epsilon >0\)  when \(|C^{\prime }|\)  is much larger than \(|B^{\prime }|\) . Indeed, if \(N\)  is the number of tuples \((c, b_1, b_2, b_3, b_4)\in C^{\prime }\times B^{\prime }\times B^{\prime }\times B^{\prime }\times B^{\prime }\)  such that \(b_1+cb_2=b_4+cb_3\) , by the Cauchy-Schwarz inequality, one has \(N\)  is at least \(|C^{\prime }||B^{\prime }|^{2+\epsilon }\) . We observe that the equation \(b_1+cb_2=b_4+cb_3\) 
is equivalent to \(b_1-b_4=c(b_2-b_3).\) 
For a fixed \(b_4\) , the above identity gives an incidence between the line defined by \(y=c(x-b_3)+b_4\) 
and the point \((b_2, b_1)\in B^{\prime }\times B^{\prime }\) . So by the Szemerédi-Trotter theorem [1]} and taking the sum over all \(b_4\in B^{\prime }\) , \(N\)  is at most \(|B^{\prime }|^3|C^{\prime }|^{2/3}+|B^{\prime }|^2|C^{\prime }|,\) 
which is smaller than \(|B^{\prime }|^{2+\epsilon }|C^{\prime }|\)  whenever \(|C^{\prime }|\ge |B^{\prime }|^{3-3\epsilon }\) .
